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arXiv:quant-ph/9612009 · Physical Review A

Alternative approach to electromagnetic field quantization in nonlinear and inhomogeneous media

L.-M. Duan, Guang‐Can Guo

Published Jul 1, 1997• 15 citations
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Abstract

A simple approach is proposed for the quantization of the electromagnetic field in nonlinear and inhomogeneous media. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg's equations we derive Maxwell's equations for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure for linear inhomogeneous media [Phys. Rev. A 43, 467 (1991)]. The explicit Hamiltonians for the second-order and third-order nonlinear quasi-steady-state processes are obtained based on this quantization procedure.

DOI
10.1103/physreva.56.925

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A simple approach is proposed for the quantization of the electromagnetic field in nonlinear and inhomogeneous media. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg's equations we derive Maxwell's equations for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure for linear inhomogeneous media [Phys. Rev. A 43, 467 (1991)]. The explicit Hamiltonians for the second-order and third-order nonlinear quasi-steady-state processes are obtained based on this quantization procedure.
The electromagnetic field is quantized in nonlinear media whose dielectric functions and nonlinear susceptibilities may vary with space. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg equations we prove that Maxwell's equations are satisfied for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure developed in Ref.[1] for linear inhomogeneous media. The general formalism is illustrated by applications to quantization of the second-order and third-order nonlinear quasi-steady-state processes.
An introduction to the quantum theory of nonlinear optics This article is provides an introduction to the quantum theory of optics in nonlinear dielectric media. We begin with a short summary of the classical theory of nonlinear optics, that is nonlinear optics done with classical fields. We then discuss the canonical formalism for fields and its quantization. This is applied to quantizing the electromagnetic field in free space. The definition of a nonclassical state of the electromagnetic field is presented, and several examples are examined. This is followed by a brief introduction to entanglement in the context of field modes. The next task is the quantization of the electromagnetic field in an inhomogeneous, linear dielectric medium. Before going on to field quantization in nonlinear media, we discuss a number of commonly employed phenomenological models for quantum nonlinear optical processes. We then quantize the field in both nondispersive and dispersive nonlinear media. Flaws in the most commonly used methods of accomplishing this task are pointed out and discussed. Once the quantization has been completed, it is used to study a multimode theory of parametric down conversion and the propagation of quantum solitons.
In this work, quantization of electromagnetic fields in a general anisotropic inhomogeneous media is considered. First, the classical Hamiltonian is derived. Once the Hamiltonian is derived, with it the classical equations of motion follow. Next, the fields are elevated to become quantum operators, for which conjugate pairs are endowed with commutators. Since as previously shown, such commutators induce functional derivatives, the quantum equations of motion can be easily derived. Next, the mode decomposition of the fields is demonstrated. Upon substitution into the classical Hamiltonian, each mode then behaves like a simple harmonic oscillator. These oscillators can then be elevated to become quantum harmonic oscillators. Both the traveling wave modes and the standing wave modes are discussed. Furthermore, the longitudinal modes are described. Finally, the case of impressed sources are considered, with the corresponding Hamiltonian and field quantization. The Green's functions for the linear system are presented. Finally, the quantum Maxwell's equations are derived. A short discussion on field-atom interaction is given.
Recently two methods have been developed for the quantization of the electromagnetic field in general dispersing and absorbing linear dielectrics. The first is based upon the introduction of a quantum Langevin current in Maxwell's equations [T. Gruner and D.-G. Welsch, Phys. Rev. A 53, 1818 (1996); Ho Trung Dung, L. Kn\"oll, and D.-G. Welsch, Phys. Rev. A 57, 3931 (1998); S. Scheel, L. Kn\"oll, and D.-G. Welsch, Phys. Rev. A 58, 700 (1998)], whereas the second makes use of a set of auxiliary fields, followed by a canonical quantization procedure [A. Tip, Phys. Rev. A 57, 4818 (1998)]. We show that both approaches are equivalent.
A technique of canonical quantization in a general dispersive nonlinear dielectric medium is presented. The medium can be inhomogeneous and anisotropic. The fields are expanded in a slowly varying envelope approximation to allow quantization. An arbitrary number of envelopes is included, assuming lossless propagation in each relevant frequency band. The resulting Lagrangian and Hamiltonian agree with known propagation equations and expressions for the dispersive energy. The central result of the theory is an expansion of the quantum Hamiltonian in terms of annihilation and creation operators corresponding to group-velocity photon-polariton excitations in the dielectric.